0
dict | 1
dict |
---|---|
{
"from": "human",
"value": "14. Two circles $C_{1}$ and $C_{2}$ with centers $A$ and $B$ are externally tangent at $T$. Let $B D$ be a segment tangent to $C_{1}$ at $D$ and let $T C$ be the segment tangent to both at $T$ with $C \\in B D$. If $A T$ is 80 units long and $B T$ is 90 units long, what is the length of $C D$?"
} | {
"from": "assistant",
"ground_truth": {
"value": "48"
}
} |
{
"from": "human",
"value": "3B. A tower made of cards requires 2 cards for one level, 7 cards for two levels, and 15 cards for a tower of three levels (see diagram). How many cards are needed for a tower of $n$ levels?"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{n(3n+1)}{2}"
}
} |
{
"from": "human",
"value": "2. (3 points) In a puddle, there live three types of amoebas: red, blue, and yellow. From time to time, any two amoebas of different types can merge into one amoeba of the third type. It is known that in the morning, there were 26 red, 31 blue, and 16 yellow amoebas in the puddle, and by evening, only one amoeba remained. What color is it?"
} | {
"from": "assistant",
"ground_truth": {
"value": "blue"
}
} |
{
"from": "human",
"value": "24. [12] Find the number of subsets $S$ of $\\{1,2, \\ldots 6\\}$ satisfying the following conditions:\n- $S$ is non-empty.\n- No subset of $S$ has the property that the sum of its elements is 10 ."
} | {
"from": "assistant",
"ground_truth": {
"value": "34"
}
} |
{
"from": "human",
"value": "PΓ©ter, RezsΕ, SΓ‘ndor, and TamΓ‘s together have 6 drawing triangles. In how many ways is this possible? (It is not certain that each of them brought a drawing triangle, for example, P-, R-, S-, T- could have $2,0,3,1$ respectively.)"
} | {
"from": "assistant",
"ground_truth": {
"value": "84"
}
} |
{
"from": "human",
"value": "1. Given the set of real numbers $\\{1,2,3, x\\}$, the maximum element of the set is equal to the sum of all elements in the set, then the value of $x$ is $\\qquad$ ."
} | {
"from": "assistant",
"ground_truth": {
"value": "-3"
}
} |
{
"from": "human",
"value": "14. The function $f(x)$ defined on the interval $[1,2017]$ satisfies $f(1)=f(2017)$, and for any $x, y \\in [1,2017]$, it holds that $|f(x)-f(y)| \\leqslant 2|x-y|$. If the real number $m$ satisfies that for any $x, y \\in [1,2017]$, it holds that $|f(x)-f(y)| \\leqslant m$, find the minimum value of $m$."
} | {
"from": "assistant",
"ground_truth": {
"value": "2016"
}
} |
{
"from": "human",
"value": "4. At a rectangular table, eight chairs are placed, four on one side and four opposite them on the other side. In how many ways can eight friends be seated at this table so that Ana and Bane sit opposite each other, and Vesna and Goran sit next to each other? (It is known that all friends have distinct names.)\n\n"
} | {
"from": "assistant",
"ground_truth": {
"value": "1152"
}
} |
{
"from": "human",
"value": "In one American company, every employee is either a Democrat or a Republican. After one Republican decided to become a Democrat, the number of each became equal. Then, three more Republicans decided to become Democrats, and the number of Democrats became twice the number of Republicans. How many employees are there in this company?"
} | {
"from": "assistant",
"ground_truth": {
"value": "18"
}
} |
{
"from": "human",
"value": "10. Two students, A and B, are selecting courses from five subjects. They have exactly one course in common, and A selects more courses than B. The number of ways A and B can select courses to meet the above conditions is $\\qquad$"
} | {
"from": "assistant",
"ground_truth": {
"value": "155"
}
} |
{
"from": "human",
"value": "8.3. In the tetrahedron $K L M N$, it is known that $K L = M N = 4$, $K M = L N = 5$, $K N = M L = 6$. Points $P, Q, R, S$ are the centers of the inscribed circles of triangles $K L M, K L N$, $K M N$ and $L M N$. Find the volume of the tetrahedron $P Q R S$. Round the answer to the nearest hundredth if necessary."
} | {
"from": "assistant",
"ground_truth": {
"value": "0.29"
}
} |
{
"from": "human",
"value": "8. Given that $f(x)$ is a function defined on $\\mathbf{R}$, and satisfies $f(x+2)[1-f(x)]=1+f(x), f(1)=$ 9997, then the value of $f(2009)$ is $\\qquad$ ."
} | {
"from": "assistant",
"ground_truth": {
"value": "9997"
}
} |
{
"from": "human",
"value": "2. The sides of a convex quadrilateral are equal to $6, 7, 8, 9$ in some order. It is known that a circle can be inscribed in this quadrilateral and a circle can be circumscribed around it. Find the area of the quadrilateral."
} | {
"from": "assistant",
"ground_truth": {
"value": "12\\sqrt{21}"
}
} |
{
"from": "human",
"value": "3. Let $N$ be a regular nonagon, $O$ the center of its circumscribed circle, $P Q$ and $Q R$ two consecutive sides of $N$, $A$ the midpoint of $P Q$, and $B$ the midpoint of the radius perpendicular to $Q R$. Find the angle between $A O$ and $A B$."
} | {
"from": "assistant",
"ground_truth": {
"value": "30"
}
} |
{
"from": "human",
"value": "68. A large group of tourists set off on an international trip. Out of them, 28 people speak English, 13 speak French, 10 speak German, 8 speak both English and French, 6 speak both English and German, 5 speak both French and German, 2 speak all three languages, and 41 people do not speak any of the three languages. How many tourists are in the group"
} | {
"from": "assistant",
"ground_truth": {
"value": "75"
}
} |
{
"from": "human",
"value": "3. In the number $2016 * * * * 02 * *$, each of the 6 asterisks needs to be replaced with any of the digits $0,2,4,5,7,9$ (digits can be repeated) so that the resulting 12-digit number is divisible by 15. In how many ways can this be done?"
} | {
"from": "assistant",
"ground_truth": {
"value": "5184"
}
} |
{
"from": "human",
"value": "G1.4 When 491 is divided by a two-digit integer, the remainder is 59 . Find this two-digit integer. Let the number be $10 x+y$, where $0<x \\leq 9,0 \\leq y \\leq 9$."
} | {
"from": "assistant",
"ground_truth": {
"value": "72"
}
} |
{
"from": "human",
"value": "4. Inside an angle of $30^{\\circ}$ with vertex $A$, a point $K$ is chosen, the distances from which to the sides of the angle are 1 and 2. Through point $K$, all possible lines are drawn, intersecting the sides of the angle. Find the minimum area of the triangle cut off by the line from the angle."
} | {
"from": "assistant",
"ground_truth": {
"value": "8"
}
} |
{
"from": "human",
"value": "4. Find the value of the expression $\\frac{1}{a b}+\\frac{1}{b c}+\\frac{1}{a c}$, if it is known that $a, b, c$ are three different real numbers satisfying the conditions $a^{3}-2020 a^{2}+1010=0, b^{3}-2020 b^{2}+1010=0, \\quad c^{3}-2020 c^{2}+1020=0$."
} | {
"from": "assistant",
"ground_truth": {
"value": "-2"
}
} |
{
"from": "human",
"value": "3. (7 points) Four mole burrows $A, B, C, D$ are connected sequentially by three tunnels. Each minute, the mole runs through a tunnel to one of the adjacent burrows. In how many ways can the mole get from burrow $A$ to $C$ in 22 minutes?\n\n"
} | {
"from": "assistant",
"ground_truth": {
"value": "17711"
}
} |
{
"from": "human",
"value": "1. (16 points) There are two circles: one with center at point $A$ and radius 5, and another with center at point $B$ and radius 15. Their common internal tangent touches the circles at points $C$ and $D$ respectively. Lines $A B$ and $C D$ intersect at point $E$. Find $C D$, if $B E=39$."
} | {
"from": "assistant",
"ground_truth": {
"value": "48"
}
} |
{
"from": "human",
"value": "24th EΓΆtvΓΆs 1917 Problem 2 A square has 10s digit 7. What is its units digit?"
} | {
"from": "assistant",
"ground_truth": {
"value": "6"
}
} |
{
"from": "human",
"value": "The distinct prime factors of 18 are 2 and 3 . What is the sum of all of the distinct prime factors of $4446 ?$"
} | {
"from": "assistant",
"ground_truth": {
"value": "37"
}
} |
{
"from": "human",
"value": "Mom gave Vasya money for 30 pencils. It turned out that the pencil factory was running a promotional campaign in the store: in exchange for a receipt for a set of 20 pencils, they return $25\\%$ of the set's cost, and for a receipt for a set of 5 pencils, they return $10\\%$. What is the maximum number of pencils Vasya can buy?"
} | {
"from": "assistant",
"ground_truth": {
"value": "36"
}
} |
{
"from": "human",
"value": "What is the smallest two-digit positive integer that is a multiple of 6 but is not a multiple of 4 ?"
} | {
"from": "assistant",
"ground_truth": {
"value": "18"
}
} |
{
"from": "human",
"value": "Suppose that the roots of the quadratic $x^2 + ax + b$ are $\\alpha$ and $\\beta$. Then $\\alpha^3$ and $\\beta^3$ are the roots of some quadratic $x^2 + cx + d$. Find $c$ in terms of $a$ and $b$.\n"
} | {
"from": "assistant",
"ground_truth": {
"value": "a^3 - 3ab"
}
} |
{
"from": "human",
"value": "8.4. On a chessboard, the centers of some cells are marked in such a way that no triangle with marked vertices is a right triangle. What is the maximum number of points that could have been marked?"
} | {
"from": "assistant",
"ground_truth": {
"value": "14"
}
} |
{
"from": "human",
"value": "Suppose you have three children and 40 pieces of candy. How many ways are there to distribute the candy such that each child gets more than one but fewer than 20 pieces?"
} | {
"from": "assistant",
"ground_truth": {
"value": "171"
}
} |
{
"from": "human",
"value": "8. The smallest positive period of the function $f(x)=|\\sin x|-|\\cos x|$ is\n\nδΏηδΊζΊζζ¬ηζ’θ‘εζ ΌεΌγ"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\pi"
}
} |
{
"from": "human",
"value": "Example 4 Given that $\\alpha^{2005}+\\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\\alpha+\\beta, \\alpha \\beta$. Find the sum of the coefficients of this polynomial.\n(2005 China Western Olympiad)"
} | {
"from": "assistant",
"ground_truth": {
"value": "1"
}
} |
{
"from": "human",
"value": "3.303. $1-\\sin ^{2} \\alpha-\\sin ^{2} \\beta+2 \\sin \\alpha \\sin \\beta \\cos (\\alpha-\\beta)$."
} | {
"from": "assistant",
"ground_truth": {
"value": "\\cos^{2}(\\alpha-\\beta)"
}
} |
{
"from": "human",
"value": "20th APMC 1997 Problem 3 The 97 numbers 49/1, 49/2, 49/3, ... , 49/97 are written on a blackboard. We repeatedly pick two numbers a, b on the board and replace them by 2ab - a - b + 1 until only one number remains. What are the possible values of the final number?"
} | {
"from": "assistant",
"ground_truth": {
"value": "1"
}
} |
{
"from": "human",
"value": "27. N8 (IRN) Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$ th power. What is the largest possible number of elements in $A$?"
} | {
"from": "assistant",
"ground_truth": {
"value": "(p-1)^2"
}
} |
{
"from": "human",
"value": "15. When classmates asked Kolya about the number of the apartment his family received in the new building, Kolya, a math enthusiast, replied: βOur apartment number is a three-digit number that is a perfect cube. If you swap the hundreds and units digits, you get a prime number.β What is Kolya's apartment number?"
} | {
"from": "assistant",
"ground_truth": {
"value": "125"
}
} |
{
"from": "human",
"value": "7. For any positive integer $n$, let $f_{1}(n)$ denote the square of the sum of the digits plus $r+1$, where $r$ is the integer satisfying $n=3 q+$ $r(q$ is an integer $), 0 \\leqslant r \\leqslant 3$. For $k \\geqslant 2$, let $f_{k}(n)=f_{1}\\left[f_{k-1}(n)\\right]$, then $f_{2001}(2345)=$ $\\qquad$ ,$f_{2001}(2001)=$ $\\qquad$ ."
} | {
"from": "assistant",
"ground_truth": {
"value": "10"
}
} |
{
"from": "human",
"value": "4. (7 points) When copying a problem, Lin Lin mistakenly wrote a repeating decimal as 0.123456. If the digits are correct but the dots indicating the repeating section were omitted, there are $\\qquad$ possible original decimals."
} | {
"from": "assistant",
"ground_truth": {
"value": "6"
}
} |
{
"from": "human",
"value": "3. The equation $x^{2}+a x+3=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,\n\n$$\nx_{1}^{3}-\\frac{99}{2 x_{2}^{2}}=x_{2}^{3}-\\frac{99}{2 x_{1}^{2}}\n$$\n\nFind all possible values of $a$."
} | {
"from": "assistant",
"ground_truth": {
"value": "-6"
}
} |
{
"from": "human",
"value": "13. Salvo-control. The artillery system of a missile cruiser fires at a target. If the target is not hit, the system automatically makes a second shot at the same target. A third shot at this target is not made. The cruiser has 10 missiles on board. During exercises, it conducts firings at several targets under identical conditions. The probability of hitting with each single shot is $p$.\n\na) (from 8th grade, 2 points). If there are only five targets, what is the probability that exactly three unused missiles will remain after firing?\n\nb) (from 9th grade, 3 points). Find the expected number of targets hit if there are nine targets in total."
} | {
"from": "assistant",
"ground_truth": {
"value": "10p-p^{10}"
}
} |
{
"from": "human",
"value": "2. Mikhail and Martin live at the same distance from the school and started for the school at the same time. Mikhail's step is $10 \\%$ shorter than Martin's, but Mikhail took $10 \\%$ more steps than Martin in the same time. Who arrived at the school first?"
} | {
"from": "assistant",
"ground_truth": {
"value": "Martin"
}
} |
{
"from": "human",
"value": "5. Given a regular tetrahedron $S-ABC$ with the base being an equilateral triangle of side length 1, and the side edges being 2. A section through $AB$ divides the volume of the tetrahedron into two equal parts. Then the cosine of the plane angle of the dihedral angle formed by the section and the base is $\\qquad$"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{2\\sqrt{15}}{15}"
}
} |
{
"from": "human",
"value": "3. Mowgli asked five monkeys to bring him some nuts. The monkeys collected an equal number of nuts and carried them to Mowgli. On the way, they quarreled, and each monkey threw one nut at each of the others. As a result, they brought Mowgli half as many nuts as they had collected. How many nuts did Mowgli receive? Be sure to explain your answer."
} | {
"from": "assistant",
"ground_truth": {
"value": "20"
}
} |
{
"from": "human",
"value": "7.5. On a chessboard, the centers of some cells are marked in such a way that no triangle with marked vertices is a right triangle. What is the maximum number of points that could have been marked?"
} | {
"from": "assistant",
"ground_truth": {
"value": "14"
}
} |
{
"from": "human",
"value": "Example 31 (2004 College Entrance Examination Hunan Paper) Given $a>0, b>0$, which of the following inequalities does not always hold? ( ).\nA. $(a+b)\\left(\\frac{1}{a}+\\frac{1}{b}\\right) \\geqslant 4$\nB. $a^{3}+b^{3} \\geqslant 2 a b^{2}$\nC. $a^{2}+b^{2}+2 \\geqslant 2 a+2 b$\nD. $\\sqrt{|a-b|} \\geqslant \\sqrt{a}-\\sqrt{b}$"
} | {
"from": "assistant",
"ground_truth": {
"value": "B"
}
} |
{
"from": "human",
"value": "8. Let $\\triangle A B C$ have internal angles $A, B, C$ with opposite sides $a, b, c$ respectively, and satisfy $a \\cos B-b \\cos A=\\frac{4}{5} c$, then $\\frac{\\tan A}{\\tan B}=$\n$\\qquad$ ."
} | {
"from": "assistant",
"ground_truth": {
"value": "9"
}
} |
{
"from": "human",
"value": "For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$.\r\nFind the maximum value of $ s(P)$ over all such sets $ P$."
} | {
"from": "assistant",
"ground_truth": {
"value": "7"
}
} |
{
"from": "human",
"value": "6. Cheburashka bought as many mirrors from Galina in the store as Gen bought from Shapoklyak. If Gen had bought from Galina, he would have 27 mirrors, and if Cheburashka had bought from Shapoklyak, he would have 3 mirrors. How many mirrors would Gen and Cheburashka buy together if Galina and Shapoklyak agreed and set the price of mirrors to the average of their current prices? (The average of two numbers is half of their sum, for example, for the numbers 22 and 28, the average is 25.)"
} | {
"from": "assistant",
"ground_truth": {
"value": "18"
}
} |
{
"from": "human",
"value": "Kalinin D.A.\n\nIn the room, there are 20 chairs of two colors: blue and red. On each chair sits either a knight or a liar. Knights always tell the truth, and liars always lie. Each of the seated individuals claimed that they were sitting on a blue chair. Then they somehow rearranged themselves, after which half of those seated said they were sitting on blue chairs, and the rest said they were sitting on red chairs. How many knights are now sitting on red chairs?"
} | {
"from": "assistant",
"ground_truth": {
"value": "5"
}
} |
{
"from": "human",
"value": "6. (6 points) Fill in a natural number in each of the $\\square$ and $\\triangle$ to make the equation true.\n$$\n\\square^{2}+12=\\triangle^{2} \\text {, then: } \\square+\\triangle=6 \\text {. }\n$$"
} | {
"from": "assistant",
"ground_truth": {
"value": "6"
}
} |
{
"from": "human",
"value": "1. It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?"
} | {
"from": "assistant",
"ground_truth": {
"value": "2"
}
} |
{
"from": "human",
"value": "5. A cube with a side of 5 is made up of 125 smaller cubes with a side of 1. How many small cubes does a plane perpendicular to one of the cube's diagonals and passing through its midpoint intersect?"
} | {
"from": "assistant",
"ground_truth": {
"value": "55"
}
} |
{
"from": "human",
"value": "8. A box contains 5 white balls and 5 black balls. Now, the 10 balls are taken out one by one, ensuring that after each ball is taken out, the number of black balls left in the box is not less than the number of white balls. Assuming balls of the same color are indistinguishable, there are $\\qquad$ ways to do this."
} | {
"from": "assistant",
"ground_truth": {
"value": "42"
}
} |
{
"from": "human",
"value": "Suppose you have 9 evenly spaced dots in a circle on a piece of paper. You want to draw a 9-pointed star by connecting dots around the circle without lifting your pencil, skipping the same number of dots each time.\n\nDetermine the number of different stars that can be drawn, if the regular nonagon does not count as a star.\n\n#"
} | {
"from": "assistant",
"ground_truth": {
"value": "2"
}
} |
{
"from": "human",
"value": "# Task β 6.1\n\n## Condition:\n\nGiven a triangle $\\mathrm{ABC}$, in which $\\mathrm{AB}=5$. The median $\\mathrm{BM}$ is perpendicular to the bisector $\\mathrm{AL}$. Find $\\mathrm{AC}$."
} | {
"from": "assistant",
"ground_truth": {
"value": "10"
}
} |
{
"from": "human",
"value": "5. Xiaohong and Xiaoliang play the \"Rock, Paper, Scissors\" game, agreeing that if they win, they go up three steps, and if they lose, they go down two steps. They start playing from the 9th step, and after 12 rounds (each round has a winner and a loser), Xiaohong is on the 10th step. How many times did Xiaohong win?"
} | {
"from": "assistant",
"ground_truth": {
"value": "5"
}
} |
{
"from": "human",
"value": "3. Let real numbers $a b$ satisfy\n$$\n\\begin{array}{l}\na^{2}\\left(b^{2}+1\\right)+b(b+2 a)=40, \\\\\na(b+1)+b=8 .\n\\end{array}\n$$\n\nFind the value of $\\frac{1}{a^{2}}+\\frac{1}{b^{2}}$."
} | {
"from": "assistant",
"ground_truth": {
"value": "8"
}
} |
{
"from": "human",
"value": "5. Two circles $k_{1}$, $k_{2}$ with radius $\\frac{1}{2} r$ are inscribed in a circle $k$ with radius $r$, and they touch each other. Circle $l$ touches circles $k_{1}$, $k_{2}$ externally and has an internal touch with circle $k$. Circle $m$ has an external touch with circles $k_{2}$ and $l$ and an internal touch with circle $k$. Calculate the radii of circles $l$ and $m$."
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{1}{6}r"
}
} |
{
"from": "human",
"value": "An [i]up-right path[/i] from $(a, b) \\in \\mathbb{R}^2$ to $(c, d) \\in \\mathbb{R}^2$ is a finite sequence $(x_1, y_z), \\dots, (x_k, y_k)$ of points in $ \\mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \\le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$. Two up-right paths are said to intersect if they share any point.\n\nFind the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$, $B$ is an up-right path from $(2, 0)$ to $(6, 4)$, and $A$ and $B$ do not intersect."
} | {
"from": "assistant",
"ground_truth": {
"value": "1750"
}
} |
{
"from": "human",
"value": "How many solutions does the equation\n\n$$\n\\{x\\}^{2}=\\left\\{x^{2}\\right\\}\n$$\n\nhave in the interval $[1 ; 100]$? $(\\{u\\}-a$ is the fractional part of $u$ - the difference between $u$ and the greatest integer not greater than $u$.)"
} | {
"from": "assistant",
"ground_truth": {
"value": "9901"
}
} |
{
"from": "human",
"value": "1. [2] Two cars are driving directly towards each other such that one is twice as fast as the other. The distance between their starting points is 4 miles. When the two cars meet, how many miles is the faster car from its starting point?"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{8}{3}"
}
} |
{
"from": "human",
"value": "[ Invariants $]$\n\nOn the board, the numbers $1,2, \\ldots, 20$ are written. It is allowed to erase any two numbers $a$ and $b$ and replace them with the number $a b + a + b$.\n\nWhat number can remain on the board after 19 such operations?"
} | {
"from": "assistant",
"ground_truth": {
"value": "21!-1"
}
} |
{
"from": "human",
"value": "16. A2 (RUS) The numbers from 1 to $n^{2}$ are randomly arranged in the cells of a $n \\times n$ square $(n \\geq 2)$. For any pair of numbers situated in the same row or in the same column, the ratio of the greater number to the smaller one is calculated.\nLet us call the characteristic of the arrangement the smallest of these $n^{2}(n-1)$ fractions. What is the highest possible value of the characteristic?"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{n+1}{n}"
}
} |
{
"from": "human",
"value": "[ Algebraic problems on the triangle inequality ]\n\nIn a triangle, two sides are equal to 3.14 and 0.67. Find the third side, given that its length is an integer.\n\n#"
} | {
"from": "assistant",
"ground_truth": {
"value": "3"
}
} |
{
"from": "human",
"value": "5. Given the equation of $\\odot O$ is $x^{2}+y^{2}=4$, and the equation of $\\odot M$ is\n$$\n(x-5 \\cos \\theta)^{2}+(y-5 \\sin \\theta)^{2}=1(\\theta \\in \\mathbf{R}),\n$$\n\na line is drawn from any point $P$ on $\\odot M$ to $\\odot O$, forming two tangents $P E$ and $P F$, with the points of tangency being $E$ and $F$. Then the minimum value of $\\overrightarrow{P E} \\cdot \\overrightarrow{P F}$ is"
} | {
"from": "assistant",
"ground_truth": {
"value": "6"
}
} |
{
"from": "human",
"value": "11. Given that vectors $\\alpha, \\beta$ are two mutually perpendicular unit vectors in a plane, and\n$$\n(3 \\boldsymbol{\\alpha}-\\boldsymbol{\\gamma}) \\cdot(4 \\boldsymbol{\\beta}-\\boldsymbol{\\gamma})=0 .\n$$\n\nThen the maximum value of $|\\boldsymbol{\\gamma}|$ is $\\qquad$"
} | {
"from": "assistant",
"ground_truth": {
"value": "5"
}
} |
{
"from": "human",
"value": "37. There are two-digit numbers where the sum of the tens digit and the units digit is 8. There are $\\qquad$ such two-digit numbers."
} | {
"from": "assistant",
"ground_truth": {
"value": "8"
}
} |
{
"from": "human",
"value": "18. Given that $a, b, c$ are non-zero rational numbers, and satisfy $a b^{2}=\\frac{c}{a}-b$, then $\\left(\\frac{a^{2} b^{2}}{c^{2}}-\\frac{2}{c}+\\frac{1}{a^{2} b^{2}}+\\frac{2 a b}{c^{2}}-\\frac{2}{a b c}\\right) \\div\\left(\\frac{2}{a b}-\\frac{2 a b}{c}\\right) \\div \\frac{101}{c}=$ $\\qquad$"
} | {
"from": "assistant",
"ground_truth": {
"value": "-\\frac{1}{202}"
}
} |
{
"from": "human",
"value": "A [i]strip[/i] is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\\circ$ be the measure (in degrees) of one interior angle of $P$. What is the greatest possible value of the number $x$?"
} | {
"from": "assistant",
"ground_truth": {
"value": "150^\\circ"
}
} |
{
"from": "human",
"value": "3.4. The angle at the base of an acute isosceles triangle \\(ABC (AB = BC)\\) is \\(\\alpha\\). In what ratio, counting from vertex \\(A\\), does the height \\(BD\\) divide the height \\(AE\\) ?"
} | {
"from": "assistant",
"ground_truth": {
"value": "-\\frac{1}{\\cos2\\alpha}"
}
} |
{
"from": "human",
"value": "8. Reviews (from 7th grade. 1 point). Angry reviews about the work of an online store are left by $80 \\%$ of dissatisfied customers (those who were poorly served in the store). Only $15 \\%$ of satisfied customers leave positive reviews. A certain online store has received 60 angry and 20 positive reviews. Using this statistics, estimate the probability that the next customer will be satisfied with the service in this online store[^0]\n\n"
} | {
"from": "assistant",
"ground_truth": {
"value": "0.64"
}
} |
{
"from": "human",
"value": "7. $n^{2}(n \\geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns\n$$\n\\begin{array}{l}\na_{11} a_{12} a_{13} a_{14} \\cdots a_{1 n} \\\\\na_{21} a_{22} a_{23} a_{24} \\cdots a_{2 n} \\\\\na_{31} a_{32} a_{33} a_{34} \\cdots a_{3 n} \\\\\n\\cdots \\cdots \\cdots \\cdots \\cdots \\\\\na_{n 1} a_{n 2} a_{n 3} a_{n 4} \\cdots a_{n n}\n\\end{array}\n$$\n\nwhere the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\\frac{1}{8}, a_{43}=\\frac{3}{16}$, find $a_{11}+a_{22}+a_{23}+\\cdots+a_{n n}$."
} | {
"from": "assistant",
"ground_truth": {
"value": "2-\\frac{1}{2^{n-1}}-\\frac{n}{2^{n}}"
}
} |
{
"from": "human",
"value": "28 Three sides $O A B, O A C$ and $O B C$ of a tetrahedron $O A B C$ are right-angled triangles, i.e. $\\angle A O B=\\angle A O C=\\angle B O C=90^{\\circ}$. Given that $O A=7, O B=2$ and $O C=6$, find the value of\n(Area of $\\triangle O A B)^{2}+(\\text { Area of } \\triangle O A C)^{2}+(\\text { Area of } \\triangle O B C)^{2}+(\\text { Area of } \\triangle A B C)^{2}$."
} | {
"from": "assistant",
"ground_truth": {
"value": "1052"
}
} |
{
"from": "human",
"value": "4. Calculate $\\lim _{x \\rightarrow 1} \\frac{\\sqrt{\\operatorname{arctg} x}-\\sqrt{\\arccos \\left(\\frac{\\sqrt{2}}{2} x\\right)}}{\\sqrt[3]{1+\\ln [x(2 x-1)]}-\\sqrt[3]{1+\\ln \\left(x^{3}-4 x+4\\right)}}$.\n\nIulian Danielescu, BrΔila"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{9}{8\\sqrt{\\pi}}"
}
} |
{
"from": "human",
"value": "Example 2 $x$ is a positive real number, find the minimum value of the function $y=x^{2}+x+\\frac{3}{x}$."
} | {
"from": "assistant",
"ground_truth": {
"value": "5"
}
} |
{
"from": "human",
"value": "1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=6000$, if $f(0)=1$ and for any $x$ the equality $f(x+3)=f(x)+2 x+3$ holds."
} | {
"from": "assistant",
"ground_truth": {
"value": "12000001"
}
} |
{
"from": "human",
"value": "334. A machine manufactures balls. A ball is considered suitable if the deviation $X$ of the ball's diameter from the design size in absolute value is less than 0.7 mm. Assuming that the random variable $X$ is normally distributed with a standard deviation $\\sigma=0.4$ mm, find how many suitable balls on average will be among a hundred manufactured."
} | {
"from": "assistant",
"ground_truth": {
"value": "92"
}
} |
{
"from": "human",
"value": "1. Given two groups of numerical sequences, each consisting of 15 arithmetic progressions containing 10 terms each. The first terms of the progressions in the first group are $1,2,3, \\ldots, 15$, and their differences are respectively $2,4,6, \\ldots, 30$. The second group of progressions has the same first terms $1,2,3, \\ldots, 15$, but the differences are respectively $1,3,5, \\ldots, 29$. Find the ratio of the sum of all elements of the first group to the sum of all elements of the second group.\n\n(12 points)"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{160}{151}"
}
} |
{
"from": "human",
"value": "1. Find the sum of all positive integers whose largest proper divisor is 55 . (A proper divisor of $n$ is divisor that is strictly less than $n$.)\nProposed by: Michael Tang"
} | {
"from": "assistant",
"ground_truth": {
"value": "550"
}
} |
{
"from": "human",
"value": "1. A natural number has a digit sum of 2013. The next number has a smaller digit sum and is not divisible by 4. What is the digit sum of the next natural number."
} | {
"from": "assistant",
"ground_truth": {
"value": "2005"
}
} |
{
"from": "human",
"value": "4. Use $1,2,3,4,5,6$ to form a six-digit number without repeating digits. By arbitrarily extracting any two adjacent digits, 5 different two-digit numbers can be obtained. The maximum sum of these 5 two-digit numbers is $\\qquad$ ."
} | {
"from": "assistant",
"ground_truth": {
"value": "219"
}
} |
{
"from": "human",
"value": "8. (6 points) Given that $\\frac{x}{9}, \\frac{y}{15}, \\frac{z}{14}$ are all simplest proper fractions, and their product is $\\frac{1}{6}$, then $x+y+z=$ $\\qquad$"
} | {
"from": "assistant",
"ground_truth": {
"value": "21"
}
} |
{
"from": "human",
"value": "Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$-meter portion of the wire, there is an $\\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its closest neighbor?"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{1}{10^{10}}"
}
} |
{
"from": "human",
"value": "1. For what smallest natural $k$ does the number 2016 $\\cdot$ 20162016 $\\cdot$ 201620162016 $\\cdot$... 20162016...2016( $k$ factors) divide without remainder by $3^{67}$?"
} | {
"from": "assistant",
"ground_truth": {
"value": "27"
}
} |
{
"from": "human",
"value": "In how many ways can two different squares be chosen from an $8 \\times 8$ chessboard such that the midpoint of the segment connecting their centers also coincides with the center of a square?"
} | {
"from": "assistant",
"ground_truth": {
"value": "480"
}
} |
{
"from": "human",
"value": "$\\mathbf{5 . 2} 020$ balls are placed into 2020 different boxes, with each ball having an equal probability of landing in any box, and the probabilities of different balls landing in the boxes are independent of each other. The expected number of empty boxes is $\\qquad$"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{2019^{2020}}{2020^{2019}}"
}
} |
{
"from": "human",
"value": "Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table."
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{1}{2013}"
}
} |
{
"from": "human",
"value": "19.1.9 * Find all integers $n$ such that $n^{4}+6 n^{3}+11 n^{2}+3 n+31$ is a perfect square."
} | {
"from": "assistant",
"ground_truth": {
"value": "10"
}
} |
{
"from": "human",
"value": "In the future, each country in the world produces its Olympic athletes via cloning and strict training\nprograms. Therefore, in the fi\fnals of the 200 m free, there are two indistinguishable athletes from each\nof the four countries. How many ways are there to arrange them into eight lanes?"
} | {
"from": "assistant",
"ground_truth": {
"value": "2520"
}
} |
{
"from": "human",
"value": "99. There are 6 identical-looking coins, but 4 are genuine, of the same weight, while 2 are counterfeit, lighter, and also weigh the same. What is the minimum number of weighings on a balance scale without weights that are needed to find both counterfeit coins?"
} | {
"from": "assistant",
"ground_truth": {
"value": "3"
}
} |
{
"from": "human",
"value": "9. Arrange the positive integers (excluding 0) that are neither perfect squares nor perfect cubes in ascending order, resulting in 2, $3,5,6,7,10, \\cdots \\cdots$, what is the 1000th number in this sequence?"
} | {
"from": "assistant",
"ground_truth": {
"value": "1039"
}
} |
{
"from": "human",
"value": "8. Given $\\theta=\\arctan \\frac{5}{12}$. Then, the principal value of the argument of the complex number $z=\\frac{\\cos 2 \\theta+\\mathrm{i} \\sin 2 \\theta}{239+\\mathrm{i}}$ is $\\qquad$"
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{\\pi}{4}"
}
} |
{
"from": "human",
"value": "3. In triangle $\\triangle A B C$, the base $A B$ of which lies on the x-axis, altitudes $A M, B N$ and $C K$ are drawn. Find the length of the base $A B$, if the coordinates of points $M(2,2)$ and $N(4,4)$ are known."
} | {
"from": "assistant",
"ground_truth": {
"value": "4\\sqrt{5}"
}
} |
{
"from": "human",
"value": "23. (5 points)\nThere are some natural numbers, like $121$ and $2442$, which read the same from left to right as from right to left. We call such numbers palindromic numbers. The difference between the smallest four-digit palindromic number and the smallest three-digit palindromic number is"
} | {
"from": "assistant",
"ground_truth": {
"value": "900"
}
} |
{
"from": "human",
"value": "6. (8 points) By expanding the expression $(1+\\sqrt{7})^{211}$ using the binomial theorem, we obtain terms of the form $C_{211}^{k}(\\sqrt{7})^{k}$. Find the value of $k$ for which such a term attains its maximum value."
} | {
"from": "assistant",
"ground_truth": {
"value": "153"
}
} |
{
"from": "human",
"value": "Example 3. Find $\\lim _{x \\rightarrow 4} \\frac{x^{2}-6 x+8}{x-4}$."
} | {
"from": "assistant",
"ground_truth": {
"value": "2"
}
} |
{
"from": "human",
"value": "8. Evaluate $\\frac{1}{3}+\\frac{1}{15}+\\frac{1}{35}+\\frac{1}{63}+\\frac{1}{99}+\\frac{1}{143}+\\frac{1}{195}$."
} | {
"from": "assistant",
"ground_truth": {
"value": "\\frac{7}{15}"
}
} |
{
"from": "human",
"value": "11. (3 points) Among the 100 natural numbers from 1 to 100, the sum of all numbers that cannot be divided by 6 is"
} | {
"from": "assistant",
"ground_truth": {
"value": "4234"
}
} |
{
"from": "human",
"value": "7.1. Mihai drew several right-angled triangles on the board, such that any two triangles do not have common points. In each of these triangles, Mihai drew the following important lines (segments): all angle bisectors, all medians, and the altitude to the hypotenuse. Counting, the total number of important lines drawn in all the triangles is 44. Determine how many right-angled isosceles triangles Mihai drew on the board."
} | {
"from": "assistant",
"ground_truth": {
"value": "6"
}
} |
{
"from": "human",
"value": "5.9 The sequence $101, 104, 116, \\cdots$ has the general term $a_{n}=100+n^{2}$, where $n=1$, $2, 3, \\cdots$. For each $n$, let $d_{n}$ denote the greatest common divisor of $a_{n}$ and $a_{n+1}$. Find the maximum value of $d_{n}$. Here $n$ takes all positive integers.\n(3rd American Invitational Mathematics Examination, 1985)"
} | {
"from": "assistant",
"ground_truth": {
"value": "401"
}
} |
{
"from": "human",
"value": "Problem 2. Vanya thought of a two-digit number, then swapped its digits and multiplied the resulting number by itself. The result turned out to be four times larger than the number he thought of. What number did Vanya think of?"
} | {
"from": "assistant",
"ground_truth": {
"value": "81"
}
} |
{
"from": "human",
"value": "Evokimov\n\nA natural number is written on the board. If you erase the last digit (in the units place), the remaining non-zero number will be divisible by 20, and if you erase the first digit, it will be divisible by 21. What is the smallest number that can be written on the board if its second digit is not equal to 0?\n\n#"
} | {
"from": "assistant",
"ground_truth": {
"value": "1609"
}
} |
Overview π
We introduce Open-Reasoner-Zero, the first open source implementation of large-scale reasoning-oriented RL training focusing on scalability, simplicity and accessibility.
To enable broader participation in this pivotal moment we witnessed and accelerate research towards artificial general intelligence (AGI), we release our source code, parameter settings, training data, and model weights. Please refer to our paper for more insights across various model sizes.
Let the Reasoner-Zero tide rise!
Main Results π
Figure 1 | Evaluation performance of Open-Reasoner-Zero-{7B, 32B}. Evaluation performance of Open-Reasoner-Zero-{7B, 32B} on benchmarks (averaged on 16 responses) during training. Using the same base model as DeepSeek-R1-Zero-Qwen-32B, Open-Reasoner-Zero-32B achieves superior performance on AIME2024, MATH500, and GPQA Diamond benchmark-requiring only a tenth of the training steps.
Figure 2 | Train-time Scale up on Train Reward and Response Length of Open-Reasoner-Zero (ORZ) - {0.5B, 1.5B, 7B, 32B}. Train Reward and Response Length increase steadily, demonstrating consistent scalability across model sizes. Interestingly, the ORZ-32B Response Length exhibits fluctuations without negatively impacting training stability, highlighting the robustness of our minimalist recipe.
Releases π¦
[2025/03/31]
We announce a major milestone for Open-Reasoner-Zero
:
- π Updated Paper with new results.
- π Easy-to-use Training Scripts:
- ORZ-1.5B training scripts and ORZ-0.5B training scripts (main results in Figure 2).
- Minimal resource training scripts: ORZ-0.5B can be run on a single A800/H800 gpu!
- π€© Updated Curated Datasets:
- 129k data in total:
- 13k hard data mined from the above 129k data.
- used in the "annealing" stage of ORZ-32B training: AIME2024 from ~41% to ~48%!
- π€ More HF Models:
- Updated HF Models:
Open-Reasoner-Zero-7B
andOpen-Reasoner-Zero-32B
. - Released HF Models:
Open-Reasoner-Zero-1.5B
andOpen-Reasoner-Zero-0.5B
.
- Updated HF Models:
- π Full Suite of Critic Models for in-depth research:
Open-Reasoner-Zero-Critic-
{0.5B, 1.5B, 7B, 32B}.
[2025/02/18]
We release Open-Reasoner-Zero
.
As part of this release, we open-source:
- π Paper on our comprehensive analysis and insights in Reasoner-Zero training
- π€ HF Model
Open-Reasoner-Zero-7B
andOpen-Reasoner-Zero-32B
- π
Our curated 57k training data
- π Training Scripts to enjoy your own Reasoner-Zero journey!
Key Features in Codebase π
- Adopt single controller trainer design, flexible and researcher-friendly.
- Colocate training and generation in the same GPUs to maximize GPU utilization.
Getting Started π
Data
We release all of curated high-quality training data in the data
folder:
- curated 129k data:
- original 57k, collected from various sources, including AIME (up to 2023), MATH, Numina-Math collection and Tulu3 MATH.
- extended 72k, mainly cleaned from OpenR1-Math-220k.
- hard 13k, mined from the first stage of ORZ-32B training.
The details for how to collect data are described in our paper.
Installation & Training Scripts
We release our Dockerfile in docker folder to facilitate the reproducibility of our training.
To install the package, run:
pip install -e .
Start ORZ-32B PPO Training
Here are the starting commands in 16 nodes.
First on master node, run:
ray start --head
# you will see logging like:
# Next steps
# To add another node to this Ray cluster, run
# ray start --address='<master-node-ip>:<master-node-port>'
then on all other nodes, run:
ray start --address='<master-node-ip>:<master-node-port>' # <master-node-ip> and <master-node-port> are from above loggings!
finally on master node, just run:
python -m playground.orz_32b_ppo
Your training log will be shown in the master node terminal.
Start ORZ-0.5B PPO Training
You can start the ORZ-0.5B PPO training in single A800/H800 node:
python -m playground.orz_0p5b_ppo
You can even run in a single A800/H800 gpu:
python -m playground.orz_0p5b_ppo_1gpu
note: since we are not in multi-node setting, no ray start
like logics are needed.
Start ORZ-7B PPO Training
Multi-node Training on 4 nodes:
# set up for multi-node training
ray start --head # on master node
ray start --address='<master-node-ip>:<master-node-port>' # then on other nodes
# then on master node, run:
python -m playground.orz_7b_ppo
Your training log will be shown in the master node terminal.
Start ORZ-1.5B PPO Training
Multi-node Training on 2 nodes:
# set up for multi-node training
ray start --head # on master node
ray start --address='<master-node-ip>:<master-node-port>' # then on other nodes
# then on master node, run:
python -m playground.orz_1p5b_ppo
Debug Settings
In the code, we leave an environment variable DEBUG_MODE
to run in debug setting for researcher to iterate. (Thought for now, we recommend using python -m playground.orz_0p5b_ppo_1gpu
for debugging.)
The debug running command examples:
# NOTE: just for debug, not final setting!
## Debug command in a single GPU with `EleutherAI/pythia-14m`
DEBUG_MODE=True python -m playground.orz_14m_ppo_mini
## Debug command in a single node (8 GPUs) with `Qwen/Qwen2.5-7B`
DEBUG_MODE=True python -m playground.orz_7b_ppo
Acknowledgements π
- This work was supported by computing resources and valuable feedback provided by StepFun and Tsinghua University.
- Our training framework is built on OpenRLHF, vllm, DeepSpeed and ray.
- Our model is based on Qwen2.5 Series of base models, including Qwen2.5-0.5B, Qwen2.5-1.5B, Qwen2.5-7B and Qwen2.5-32B.
- We thank Project Numina, Tulu3 and OpenR1-Math-220k for their collected open sourced data.
Advertisement Time π£
We are hiring talented researchers and engineers to join our team. If you are interested in our project and would like to contribute to the reasoner scale-up all the way to AGI, please feel free to reach out to us at [email protected]
Community Discussions πΊ
We have several wechat groups to help discussions and sharing, you can scan the QR code below to join the latest group.

Citation
@misc{OpenReasonerZero2025,
title={Open-Reasoner-Zero: An Open Source Approach to Scaling Reinforcement Learning on the Base Model},
author={Jingcheng Hu and Yinmin Zhang and Qi Han and Daxin Jiang and Xiangyu Zhang, Heung-Yeung Shum},
year={2025},
howpublished={\url{https://github.com/Open-Reasoner-Zero/Open-Reasoner-Zero}},
}
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